Theorem pmtrsn 19448

Description: The value of the transposition generator function for a singleton is empty, i.e. there is no transposition for a singleton. This also holds for 𝐴 ∉ V, i.e. for the empty set {𝐴} = ∅ resulting in (pmTrsp‘∅) = ∅. (Contributed by AV, 6-Aug-2019.)

Assertion

Ref	Expression
pmtrsn	⊢ (pmTrsp‘{𝐴}) = ∅

Proof of Theorem pmtrsn

Dummy variables 𝑝 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 5381	. . 3 ⊢ {𝐴} ∈ V
2		eqid 2736	. . . 4 ⊢ (pmTrsp‘{𝐴}) = (pmTrsp‘{𝐴})
3	2	pmtrfval 19379	. . 3 ⊢ ({𝐴} ∈ V → (pmTrsp‘{𝐴}) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
4	1, 3	ax-mp 5	. 2 ⊢ (pmTrsp‘{𝐴}) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
5		eqid 2736	. . . . 5 ⊢ (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
6	5	dmmpt 6198	. . . 4 ⊢ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = {𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) ∈ V}
7		2on0 8411	. . . . . . . . 9 ⊢ 2o ≠ ∅
8		ensymb 8939	. . . . . . . . . 10 ⊢ (∅ ≈ 2o ↔ 2o ≈ ∅)
9		en0 8955	. . . . . . . . . 10 ⊢ (2o ≈ ∅ ↔ 2o = ∅)
10	8, 9	bitri 275	. . . . . . . . 9 ⊢ (∅ ≈ 2o ↔ 2o = ∅)
11	7, 10	nemtbir 3028	. . . . . . . 8 ⊢ ¬ ∅ ≈ 2o
12		snnen2o 9145	. . . . . . . 8 ⊢ ¬ {𝐴} ≈ 2o
13		0ex 5252	. . . . . . . . 9 ⊢ ∅ ∈ V
14		breq1 5101	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → (𝑦 ≈ 2o ↔ ∅ ≈ 2o))
15	14	notbid 318	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (¬ 𝑦 ≈ 2o ↔ ¬ ∅ ≈ 2o))
16		breq1 5101	. . . . . . . . . 10 ⊢ (𝑦 = {𝐴} → (𝑦 ≈ 2o ↔ {𝐴} ≈ 2o))
17	16	notbid 318	. . . . . . . . 9 ⊢ (𝑦 = {𝐴} → (¬ 𝑦 ≈ 2o ↔ ¬ {𝐴} ≈ 2o))
18	13, 1, 15, 17	ralpr 4657	. . . . . . . 8 ⊢ (∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2o ↔ (¬ ∅ ≈ 2o ∧ ¬ {𝐴} ≈ 2o))
19	11, 12, 18	mpbir2an 711	. . . . . . 7 ⊢ ∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2o
20		pwsn 4856	. . . . . . . 8 ⊢ 𝒫 {𝐴} = {∅, {𝐴}}
21	20	raleqi 3294	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2o ↔ ∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2o)
22	19, 21	mpbir 231	. . . . . 6 ⊢ ∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2o
23		rabeq0 4340	. . . . . 6 ⊢ ({𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} = ∅ ↔ ∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2o)
24	22, 23	mpbir 231	. . . . 5 ⊢ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} = ∅
25	24	rabeqi 3412	. . . 4 ⊢ {𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) ∈ V} = {𝑝 ∈ ∅ ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) ∈ V}
26		rab0 4338	. . . 4 ⊢ {𝑝 ∈ ∅ ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) ∈ V} = ∅
27	6, 25, 26	3eqtri 2763	. . 3 ⊢ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅
28		mptrel 5774	. . . 4 ⊢ Rel (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
29		reldm0 5877	. . . 4 ⊢ (Rel (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) → ((𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅ ↔ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅))
30	28, 29	ax-mp 5	. . 3 ⊢ ((𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅ ↔ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅)
31	27, 30	mpbir 231	. 2 ⊢ (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅
32	4, 31	eqtri 2759	1 ⊢ (pmTrsp‘{𝐴}) = ∅

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1541   ∈ wcel 2113  ∀wral 3051  {crab 3399  Vcvv 3440   ∖ cdif 3898  ∅c0 4285  ifcif 4479  𝒫 cpw 4554  {csn 4580  {cpr 4582  ∪ cuni 4863   class class class wbr 5098   ↦ cmpt 5179  dom cdm 5624  Rel wrel 5629  ‘cfv 6492  2_oc2o 8391   ≈ cen 8880  pmTrspcpmtr 19370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-1o 8397  df-2o 8398  df-er 8635  df-en 8884  df-pmtr 19371

This theorem is referenced by:  psgnsn  19449